THREE-DIMENSIONAL PROBLEM OF THE THEORY OF ELASTICITY STRESS IN A THICK-WALLED PRESSURE VESSEL

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THREE-DIMENSIONAL PROBLEM OF THE THEORY OF ELASTICITY STRESS IN A THICK-WALLED PRESSURE VESSEL 1.

forces acting inside. The analytical solution is known only for simple cases. In general, numerical methods are the only way to solve such tasks.

PROBLEM DESCRIPTION The goal of analysis is to determine stress distribution inside a pressure vessel made of steel which is a part of hydraulic installation.

1 THREE-DIMENSIONAL PROBLEM OF THE THEORY OF ELASTICITY STRESS IN A THICK-WALLED PRESSURE VESSEL 1. INTRODUCTION Three-dimensional problem of the theory of elasticity includes an elastic body with defined kinematic or static boundary conditions and the mass forces acting inside. The analytical solution is known only for simple cases. In general, numerical methods are the only way to solve such tasks. Numerical solution of the problem by using FEM requires a three-dimensional spatial discretization with a solid three-dimensional finite elements. 2. PROBLEM DESCRIPTION The goal of analysis is to determine stress distribution inside a pressure vessel made of steel which is a part of hydraulic installation. The vessel is loaded with internal pressure p. The vessel is attached by two flanges. The other two nozzles are free of displacements. Data: p=50mpa, E= MPa, ν=0.3 Geometric data (in millimeters) are presented below:

3. TYPICAL COURSE OF NUMERICAL ANALYSIS Taking into consideration the triple symmetry (xz, yz and zx planes), the model includes only ⅛ part of the vessel. Convenient units are: mm, N and MPa. 3.1.

Create Keypoints active Coordinate System Preprocessor> Modeling>Create>Keypoint> In Active CS: X=180, Y=1500 Preprocessor> Modeling>Create>Keypoint> In Active CS: X=180, Y=1200 Preprocessor>

2 3. TYPICAL COURSE OF NUMERICAL ANALYSIS Taking into consideration the triple symmetry (xz, yz and zx planes), the model includes only ⅛ part of the vessel. Convenient units are: mm, N and MPa Preprocessor 1. Create Keypoints active Coordinate System Preprocessor> Modeling>Create>Keypoint> In Active CS: X=180, Y=1500 Preprocessor> Modeling>Create>Keypoint> In Active CS: X=180, Y=1200 Preprocessor> Modeling>Create>Keypoint> In Active CS: X=180, Y=1100 Preprocessor> Modeling>Create>Keypoint> In Active CS: X=180, Y=0 2. Copy Keypoins on X direction (right) Copy>Keypoint No 4: DX=100 Copy>Keypoint No 3: DX=100 Copy>Keypoint No 2: DX=200 Copy>Keypoint No 1: DX= Create areas through Keypoints Create>Areas>Arbitrary> Through KPs: 3,4,8,7 Create>Areas>Arbitrary> Through KPs: 2,3,7,6 Create>Areas>Arbitrary> Through KPs: 1,2,6,5 2

4. Create Keypoints in active Coordinate System (on axis of

X=0, Y=200 Preprocessor> Modeling>Create>Keypoint> In Active CS: X=0,

Extrude areas about axis defined by two Keypoints: Operate>

3 4. Create Keypoints in active Coordinate System (on axis of revolution - Y) Preprocessor> Modeling>Create>Keypoint> In Active CS: X=0, Y=200 Preprocessor> Modeling>Create>Keypoint> In Active CS: X=0, Y=0 5. Extrude areas about axis defined by two Keypoints: Operate> Extrude>Areas>About Axis Create ¼ of cylinder: 7. Create ¼ of cylinder for flange: 8. Overlap Volumes: Operate> Booleans>Overlap>Volumes: All 3

4 9. Delete unnecessary Volumes: Delete> Volumes and Below 10. Select Element types: Preprocessor>Element Type>Add> (SOLID45 and SOLID95) 11. Define Material Properties: Preprocessor>Material Props>Material Models: Structural/Linear/Elastic/Isotropic: EX=2e5MPa, PRXY=0.3 4

Solution Define boundary conditions: 14. Define Symmetry B.C.

5 12. Define global element size: Preprocessor>Meshing> Meshing Tool> Size Controls>Global 13. Mesh Volumes: Preprocessor>Meshing> Meshing Tool> Mesh>Volumes/Hex/Sweep 3.2. Solution Define boundary conditions: 14. Define Symmetry B.C. on Areas: Solution>Define Loads> Apply>Structural>Displacement> Symmetry BC>On Areas 15. Define pressure on internal Areas: Solution>Define Loads> Apply>Structural>Pressure>On Areas 5

16. Define negative pressure on nozzle and

Apply>Structural>Pressure>On Areas -50 180

217MPa 50 MPa -50 700 2 /(1100 2-900 2 ) =

6 16. Define negative pressure on nozzle and flange areas: Solution>Define Loads> Apply>Structural>Pressure>On Areas /( ) = MPa 50 MPa /( ) = MPa 17. Select nodes on the sticking surface of the flange: 6

18. Couple DOFs (UZ) on the sticking surface of

7 18. Couple DOFs (UZ) on the sticking surface of the flange: Preprocessor>Coupling / Ceqn> Couple DOFs ATTENTION: Steps 17 and 18 should be performed for each model!!! 19. Select all entities: 20. Solve linear problem: Solution>Solve>Current LS 21. Save database with a unique name: Model_1.db 7

SY) in global cylindrical system related to cylindrical part of the

8 3.3. General postprocessor Show the results as contour maps: Show total displacements (USUM), Von Mises stress (SEQV) and stress components (SX, SY) in global cylindrical system related to cylindrical part of the model. 22. Plot Total displacements (USUM) 23. Plot Von Mises stress (SEQV) 8

9 24. Select global cylindrical CS for results presentation: 25. Plot radial stresses in global cylindrical CS (RSYS=1) 26. Plot hoop stress in global cylindrical CS (RSYS=1) 9

10 27. Define path AB along wall thickness and map radial and hoop stresses on it: B A 28. Plot graphs of radial and hoop stresses along path AB: 29. List radial and hoop stresses along path AB: 10

11 30. Define path CD along wall thickness and map radial and hoop stresses on it: C D 31. Plot graphs of radial and hoop stresses along path CD: 32. List radial and hoop stresses along path CD: 11

12 33. Define path EF along wall thickness for equivalent stress linearization: 34. Linearization of equivalent stress along path EF 35. List linearized equivalent stress along path EF 12

13 4. INTERPRETATION OF THE RESULTS. TASKS TO BE DONE Compare results of the models built with the same mesh density (ESIZE parameter see p.12) using: a) 8-noded elements (Solid45) using sweepping HEX/WEDGE option (Model 1), b) 20-noded elements (Solid95) using sweepping HEX/WEDGE option (Model 2), c) 8-noded elements (Solid45) using free meshing TETRA option (Model 3). Put the results in the table for each model: No. of nodes, No. of elements, USUM max, SEQV max, SX RSYS=1, SY RSYS=1 for points: A,B,C i D and maximum Membrane and Bending SEQV stress on path EF (step 35). Discuss the results. Model 1 Solid 45 Hex/Wed Model 2 Solid 95 Hex/Wed Model 3 Solid 45 Free No. of nodes No. of elements USUM max SEQV max Plots needed (should be archived during program session for each model) : 1) FE mesh 2) USUM(x,y) 3) SEQV(x,y) 4) SX(x,y) RSYS=1 5) SY(x,y) RSYS=1 SX A RSYS=1 SY A RSYS=1 6) Graph: SX(x,y) RSYS=1 i SY(x,y) RSYS=1 on path AB 7) Graph: SX(x,y) RSYS=1 i SY(x,y) RSYS=1 on path CD 8) Graph of linearized SEQV on path EF SX B RSYS=1 SY B RSYS=1 SX C RSYS=1 SY C RSYS=1 SX D RSYS=1 SY D RSYS=1 Raport finalny: Final report: 1) Introduction 2) Assumptions for the modeling 3) model description (solid model, mesh, boundary cond. and loads) 4) Results 5) Results in the Table 6) Discursion 7) Conclusion Max Membrane + Bending stress from Lame theorem (for inside pressure): 13

TWO-DIMENSIONAL PROBLEM OF THE THEORY OF ELASTICITY. INVESTIGATION OF STRESS CONCENTRATION FACTORS.

Ex_1_2D Plate.doc 1 TWO-DIMENSIONAL PROBLEM OF THE THEORY OF ELASTICITY. INVESTIGATION OF STRESS CONCENTRATION FACTORS. 1. INTRODUCTION Two-dimensional problem of the theory of elasticity is a particular